Place egg yolk, garlic, juice, and salt in a blender; blend until mixture is light and smooth; with motor running, gradually add oil; blend until emulsified and smooth; place in airtight container; cover; reserve in refrigerator.

While fish is cooking, pour just enough oil and water into large sauté pan to coat vegetables lightly; heat until hot; add potatoes, beet, and carrot; cook, stirring occasionally, until heated through; remove from heat; reserve (keep warm).

To serve, evenly divide vegetables among warm serving plates, arranging in attractive pattern; using a fish spatula, place a halibut fillet in center of each plate; spoon dollop of aïoli on top of each fillet; garnish with basil leaves.

What to drink: Bandol La Bastide Blanche 2009
Chef’s Note: Typically, halibut is cooked quickly—pan-seared, broiled, or baked in a very hot oven—but cooking the fish in a bath of extra-virgin olive oil, traps in the moisture as the fish comes up to temperature, resulting in a very satisfying texture.